A sphere centered at the origin rotates (direction not specified) about the $z$-axis with angular velocity $\omega$. Let $P(r, \theta, \phi)$ be any point inside the sphere. Find the direction (unit vector in spherical coordinates) in which $P$ is moving.
My thoughts:
We must find a line tangent to a sphere of radius $r$ (centered at the origin) at $P$, which is also parallel to the $xy$-plane. Then I'm stuck.
On
Solution: there is only one circle, parallel to the X-Y plane, that also contains the point P. The answer to the problem is the unit tangent vector of that circle at point P. The sign of the angular velocity determines which of the two possible directions the vector can point in (depends on whether the rotation is CW or CCW). BTW, the unit tangent vector will also be parallel to the X-Y plane.
Oh, and I almost forgot, P cannot lie on the Z axis (r cannot be zero and phi cannot be 90 deg or -90 deg). Otherwise the trivial result is zero motion.
I think the answer is just $\mathbf{\hat{\phi}}$.
Can someone please confirm?